Graduate Studies, UNL

 

Dissertations and Doctoral Documents, University of Nebraska-Lincoln, 2023–

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First Advisor

Eloísa Grifo

Degree Name

Doctor of Philosophy (Ph.D.)

Committee Members

Jack Jeffries, Mark Walker, Suna Gunther

Department

Mathematics

Date of this Version

4-21-2026

Document Type

Dissertation

Citation

A dissertation presented to the faculty of the Graduate College at the University of Nebraska in partial fulfillment of requirements for the degree Doctor of Philosophy

Major: Mathematics

Under the supervision of Professor Eloisa Grifo

Lincoln, Nebraska, May 2026

Comments

Copyright 2026, Ryan Watson. Used by permission

Abstract

Inspired by Quillen’s geometric approach to study group cohomology, Avramov introduced the theory of support varieties to study finite modules over local complete intersection rings. Geometric properties of the variety encode important homological information, and this theory has been a useful tool in studying complete intersection rings leading to many advances in local commutative algebra. By the work of several authors, this theory has now been expanded to encompass any noetherian local ring. Notably, Pollitz developed the theory of cohomological support varieties over Koszul complexes and used them to answer a question of Dwyer, Greenlees, and Iyengar regarding the structure of the derived category of complete intersections. The main results in this thesis are concerned with how these cohomological support varieties behave when restricting or extending scalars along certain local homomorphisms. As cohomological support varieties contain delicate homological information, this allows one to see how this information is transferred along those functors. Additionally, this thesis contains a method for computing cohomological support varieties that is implemented in the Macaulay2 package, ThickSubcategories currently being developed by Grifo, Letz, and Pollitz.

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